Place the chicken breasts in the bottom of the slow cooker. Add the wild rice, diced carrots, celery, onion, and minced garlic on top.
Pour in the chicken broth and sprinkle the dried thyme, salt, and black pepper over everything. Stir gently to distribute the seasonings.
Cover and cook on low for 6 to 7 hours or on high for 3 to 4 hours until the chicken is fully cooked and the wild rice is tender.
Remove the chicken breasts, shred them with two forks, and return the meat to the slow cooker.
Stir in the heavy cream and let the soup cook for 15 more minutes on low to warm through and thicken slightly.
Ladle into bowls and sprinkle with chopped fresh parsley before serving.